C2 January 2005 Q2
2. The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.
(a) Find the coordinates of the mid-point of \(AB\). (2)
Given that \(AB\) is a diameter of the circle \(C\),
(b) find an equation for \(C\). (4)
| Scheme | Marks |
|---|---|
| \(\left(\dfrac{5 + 13}{2}, \dfrac{-1 + 11}{2}\right), = (9, 5)\) | M1, A1 |
| (2) |
Notes
M1 for some use of correct formula. can be implied
Use of \(\left(\tfrac{1}{2}(x_A - x_B), \tfrac{1}{2}(y_A - y_B)\right) \to (4, 6)\) is M0A0
| Scheme | Marks |
|---|---|
| \(r^2 = (9 - 5)^2 + (5 - -1)^2\ (= 52)\) | M1 |
| Equation of circle: \((x - 9)^2 + (y - 5)^2 = 52\) | M1, A1ft A1 |
| (4) | |
| (6 marks) |
Notes
M1 attempt to find \(r\) or \(r^2\). ft their \((9, 5)\)
\(r = AB = \sqrt{208}\) is M0
2nd M1 for \((x - 9)^2 + (y - 5)^2 =\) constant. (ft their \((9, 5)\))
A1ft for \((x - 9)^2 + (y - 5)^2 =\) their \(r^2\). (ft their \((9, 5)\) and \(r^2\))
A1 for \((x - 9)^2 + (y - 5)^2 = 52\) only.